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11    REGULARIZATION       293

(b)    Show that there exists a sequence of indices (fcJ)1^J^m such that the intervals
lkj are mutually disjoint and such that

tJi.1

(We may assume that no lk is contained in the union of the Ih such that h^k. Show
that if we put Ifc= [ak, bk] and arrange the Ik so that ak^ ak + 1, then necessarily
ak<ak+it b2k-l<a2k + 1i and b2k<a2k + 2, by considering three intervals with con-
secutive indices. Deduce that the intervals I2k-i are mutually disjoint, the I2k are
mutually disjoint and that the kj may therefore be taken to be either the even indices
or the odd indices.)

(c)    Deduce from (a) and (b) that if Ea(/) is the set of x e R such that 6(f)(x) > a,
then

,2 r + 0
aJ _0

(d)   Let g be a nonnegative function in ^(A) such that (i) g(t) = g(t)\ (ii) g is
decreasing  on [0, + oo[; (iii)  \g(t)dt= 1. Prove that | (g */)(*) | < 6(f)(x) for all

x e R. (For each a > 0, let ]h(oc), h(oc)[ be the largest open interval in whichg(t) > x.
Show that


=

/ o

/. +ao

and observe that         h(a) d<x.  %.)

J o

(e)   State and prove analogous results for functions which are integrable with respect
to Haar measure on the torus T.

11. REGULARIZATION

(14.11.1)   Let (fn) be a sequence of fi-integrable functions which satisfy the
following conditions:

(a)    the sequence of integrals \\fn(x)\ dfi(x) is bounded;

(b)    the sequence of integrals I fn(x) dfi(x) tends to 1 ;

(c)    for   each   neighborhood   V   of   e,   the   sequence    of   integrals
Cy \fn(x)\ dft(x) tends to 0.

Then

(i) For each bounded continuous function g on G, the sequence (/ * g)
converges uniformly to g on every compact subset of G. If g is uniformly
continuous with respect to a right-invariant distance on G, then the sequence
(fn * 9) converges uniformly to g in G.the function 7*  /is finite almost everywhere.